Propensity Score Modeling and Evaluation
In causal inference for binary treatments, the propensity score is defined as the probability of receiving the treatment given covariates. Under the ignorability assumption, causal treatment effects can be estimated by conditioning on/adjusting for the pr
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Propensity Score Modeling and Evaluation Yeying Zhu and Lin (Laura) Lin
Abstract In causal inference for binary treatments, the propensity score is defined as the probability of receiving the treatment given covariates. Under the ignorability assumption, causal treatment effects can be estimated by conditioning on/adjusting for the propensity scores. However, in observational studies, propensity scores are unknown and need to be estimated from the observed data. Estimation of propensity scores is essential in making reliable causal inference. In this chapter, we first briefly discuss the modeling of propensity scores for a binary treatment; then we will focus on the estimation of the generalized propensity scores for categorical treatment variables with more than two levels and continuous treatment variables. We will review both parametric and nonparametric approaches for estimating the generalized propensity scores. In the end, we discuss how to evaluate the performance of different propensity score models and how to choose an optimal one among several candidate models.
1 Propensity Score Modeling for a Binary Treatment The potential outcomes framework [23] has been a popular framework for estimating causal treatment effects. An important quantity to facilitate causal inference has been the propensity score [22], defined as the probability of receiving the treatment given a set of measured covariates. In observational studies, propensity scores are unknown and need to be estimated from the observed data. Consistent estimation of propensity scores is essential in making reliable causal inference. In this section, we briefly review the modeling of propensity scores for a binary treatment variable. We first define some notations. Let Y denote the response of interest, T be the treatment variable, and X be a p-dimensional vector of baseline covariates. The data can be represented as .Yi ; Ti ; Xi /, i D 1; : : : ; n, a random sample from .Y; T; X/. In addition to the observed quantities, we further define Yi .t/ as the potential outcome
Y. Zhu () • L. (Laura) Lin Department of Statistics & Actuarial Science, University of Waterloo, Waterloo, ON, Canada e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 H. He et al. (eds.), Statistical Causal Inferences and Their Applications in Public Health Research, ICSA Book Series in Statistics, DOI 10.1007/978-3-319-41259-7_6
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Y. Zhu and L. (Laura) Lin
if subject i were assigned to treatment level t. Here, T is a random variable and t is a specific level of T. In the case of a binary treatment, let T D 1 if treated and T D 0 if untreated. The propensity score is then defined as r.X/ P.T D 1jX/. The quantities we are interested in estimating are usually the average treatment effect (ATE): ATE D EŒY.1/ Y.0/; and the average treatment effect among the treated (ATT): ATT D EŒY.1/ Y.0/jT D 1:
1.1 Parametric Approaches In the causal inference literature, propensity score for a binary treatment variable i
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